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Part V: Continuous Random Variables

Part V: Continuous Random Variables. http:// rchsbowman.wordpress.com/2009/11/29 / statistics-notes-%E2%80%93-properties-of-normal-distribution-2/. Chapter 23: Probability Density Functions. http:// divisbyzero.com/2009/12/02

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Part V: Continuous Random Variables

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  1. Part V: Continuous Random Variables http://rchsbowman.wordpress.com/2009/11/29 /statistics-notes-%E2%80%93-properties-of-normal-distribution-2/

  2. Chapter 23: Probability Density Functions http://divisbyzero.com/2009/12/02 /an-applet-illustrating-a-continuous-nowhere-differentiable-function//

  3. Comparison of Discrete vs. Continuous (Examples)

  4. Comparison of mass vs. density

  5. Example 1 (class) Let x be a continuous random variable with density: • What is P(0 ≤ X ≤ 3)? • Determine the CDF. • Graph the density. • Graph the CDF. • Using the CDF, calculate P(0 ≤ X ≤ 3), P(2.5 ≤ X ≤ 3)

  6. Example 1 (cont.)

  7. Example 2 Let X be a continuous function with CDF as follows What is the density?

  8. Comparison of CDFs

  9. Example 3 Suppose a random variable X has a density given by: Find the constant k so that this function is a valid density.

  10. Example 4 Suppose a random variable X has the following density: • Find the CDF. • Graph the density. • Graph the CDF.

  11. Example 4 (cont.)

  12. Mixed R.V. – CDF Let X denote a number selected at random from the interval (0,4), and let Y = min(X,3). Obtain the CDF of the random variable Y.

  13. Chapter 24: Joint Densities http://www.alexfb.com/cgi-bin/twiki/view/PtPhysics/WebHome

  14. Probability for two continuous r.v. http://tutorial.math.lamar.edu/Classes/CalcIII/DoubleIntegrals.aspx

  15. Example 1 (class) A man invites his fiancée to a fine hotel for a Sunday brunch. They decide to meet in the lobby of the hotel between 11:30 am and 12 noon. If they arrive a random times during this period, what is the probability that they will meet within 10 minutes? (Hint: do this geometrically)

  16. Example: FPF (Cont)

  17. Example 2 (class) Consider two electrical components, A and B, with respective lifetimes X and Y. Assume that a joint PDF of X and Y is fX,Y(x,y) = 10e-(2x+5y), x, y > 0 and fX,Y(x,y) = 0 otherwise. a) Verify that this is a legitimate density. b) What is the probability that A lasts less than 2 and B lasts less than 3? c) Determine the joint CDF. d) Determine the probability that both components are functioning at time t. e) Determine the probability that A is the first to fail. f) Determine the probability that B is the first to fail.

  18. Example 2d

  19. Example 2e

  20. Example 2e

  21. Example 3 Suppose a random variables X and Y have a joint density given by: Find the constant k so that this function is a valid density.

  22. Example 4 (class) Suppose a random variables X and Y have a joint density given by: • Verify that this is a valid joint density. • Find the joint CDF. • From the joint CDF calculated in a), determine the density (which should be what is given above).

  23. Example: Marginal density (class) A bank operates both a drive-up facility and a walk-up window. On a randomly selected day, let X = the proportion of time that the drive-up facility is in use (at least one customer is being served or waiting to be served) and Y = the proportion of time that the walk-up window is in use. The joint PDF is a) What is fX(x)? b) What is fY(y)?

  24. Example: Marginal density (homework) A nut company markets cans of deluxe mixed nuts containing almonds, cashews and peanuts. Suppose the net weight of each can is exactly 1 lb, but the weight contribution of each type of nut is random. Because the three weights sum to 1, a joint probability model for any two gives all necessary information about the weight of the third type. Let X = the weight of almonds in a selected can and Y = the weight of cashews. The joint PDF is a) What is fX(x)? b) What is fY(y)?

  25. Chapter 25: Independent Why’s everything got to be so intense with me? I’m trying to handle all this unpredictability In all probability -- Long Shot, sung by Kelly Clarkson, from the album All I ever Wanted; song written by Katy Perry, Glen Ballard, Matt Thiessen

  26. Example: Independent R.V.’s A bank operates both a drive-up facility and a walk-up window. On a randomly selected day, let X = the proportion of time that the drive-up facility is in use (at least one customer is being served or waiting to be served) and Y = the proportion of time that the walk-up window is in use. The joint PDF is Are X and Y independent?

  27. Example: Independence Consider two electrical components, A and B, with respective lifetimes X and Y with marginal shown densities below which are independent of each other. fX(x) = 2e-2x, x > 0, fY(y) = 5e-5y, y > 0 and fX(x) = fY(y) = 0 otherwise. What is fX,Y(x,y)?

  28. Example: Independent R.V.’s (homework) A nut company markets cans of deluxe mixed nuts containing almonds, cashews and peanuts. Suppose the net weight of each can is exactly 1 lb, but the weight contribution of each type of nut is random. Because the three weights sum to 1, a joint probability model for any two gives all necessary information about the weight of the third type. Let X = the weight of almonds in a selected can and Y = the weight of cashews. The joint PDF is Are X and Y independent?

  29. Chapter 26: Conditional Distributions Q : What is conditional probability?A : maybe, maybe not. http://www.goodreads.com/book/show/4914583-f-in-exams

  30. Example: Conditional PDF (class) Suppose a random variables X and Y have a joint density given by: • Calculate the conditional density of Xwhen Y = y where 0 < y < 1. • Verify that this function is a density. • What is the conditional probability that X is between -1 and 0.5 when we know that Y = 0.6. • Are X and Y independent? (Show using conditional densities.)

  31. Chapter 27: Expected values http://www.qualitydigest.com/inside/quality-insider-article /problems-skewness-and-kurtosis-part-one.html#

  32. Comparison of Expected Values

  33. Example: Expected Value (class) What is the expected value in each of the following situations: a) The following is the density of the magnitude X of a dynamic load on a bridge (in newtons) b) The train to Chicago leaves Lafayette at a random time between 8 am and 8:30 am. Let X be the departure time.

  34. Chapter 28: Functions, Variance http://quantivity.wordpress.com/2011/05/02/empirical-distribution-minimum-variance/

  35. Comparison of Functions, Variances

  36. Example: Expected Value - function (class) What is (X2) in each of the following situations: a) The following is the density of the magnitude X of a dynamic load on a bridge (in newtons) b) The train to Chicago leaves Lafayette at a random time between 8 am and 8:30 am. Let X be the departure time.

  37. Example: Variance (class) What is the variance in each of the following situations: a) The following is the density of the magnitude X of a dynamic load on a bridge (in newtons) b) The train to Chicago leaves Lafayette at a random time between 8 am and 8:30 am. Let X be the departure time.

  38. Friendly Facts about Continuous Random Variables - 1 • Theorem 28.18: Expected value of a linear sum of two or more continuous random variables: (a1X1 + … + anXn) = a1(X1) + … + an(Xn) • Theorem 28.19: Expected value of the product of functions of independent continuous random variables: (g(X)h(Y)) = (g(X))(h(Y))

  39. Friendly Facts about Continuous Random Variables - 2 • Theorem 28.21: Variances of a linear sum of two or more independent continuous random variables: Var(a1X1 + … + anXn) =Var(X1) + … + Var(Xn) • Corollary 28.22: Variance of a linear function of continuous random variables: Var(aX + b) = a2Var(X)

  40. Chapter 29: Summary and Review http://www.ux1.eiu.edu/~cfadd/1150/14Thermo/work.html

  41. Example: percentile Let x be a continuous random variable with density: • What is the 99th percentile? • What is the median?

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